Use Theorem to prove that if m and n are any positive integers and m is odd, then is divisible by m. Does the conclusion hold if m is even? Justify your answer.
Theorem 1
Sum of the First n Integers
For all integers n ≥ 1,
Proof (by mathematical induction):
Let the property P(n) be the equation
Show that P(1) is true:
To establish P(1), we must show that
But the left-hand side of this equation is 1 and the right-hand side is
also. Hence P(1) is true.
Show that for all integers k≥ 1, if P(k) is true then P(k + 1) is also true:
[Suppose that P(k) is true for a particular but arbitrarily chosen integer k ≥ 1.
That is:] Suppose that k is any integer with k ≥ 1 such that
[We must show that P(k + 1) is true. That is:] We must show that
or, equivalently, that
[We will show that the left-hand side and the right-hand side of P(k + 1) are equal to the same quantity and thus are equal to each other.]
The left-hand side of P(k + 1) is
And the right-hand side of P(k + 1) is
Thus the two sides of P(k + 1) are equal to the same quantity and so they are equal to each other. Therefore the equation P(k + 1) is true [as was to be shown].
[Since we have proved both the basis step and the inductive step, we conclude that the
theorem is true.]
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